Expert Answer to "Nonlinear first oder ODEI want to solve the..."

Madilynn Avery

Answered question

2022-04-21

Nonlinear first oder ODE
I want to solve the following nonlinear ODE

y^{'} - \frac{y}{x} + \frac{3}{y} (\frac{1}{2} - x) + 1 = 0

Answer & Explanation

Yaretzi Odom

Beginner2022-04-22Added 16 answers

Step 1
Consider the homogeneous equation

y^{'} - \frac{y}{x} + \frac{3}{y} (\frac{1}{2} - x) = 0

which does not present any problem. Let

y = \sqrt{z}

which gives

\frac{x (z^{'} - 6 x + 3) - 2 z}{2 x \sqrt{z}} = 0 \Rightarrow z = C x^{2} + 6 x^{2} \log (x) + 3 x

Back to y and including the 1 and the variation of parameters, we end with

\frac{x^{2} C^{'}}{\sqrt{C x^{2} + 6 x^{2} \log (x) + 3 x}} + 2 = 0

which is obviously more than problematic.

RormFrure6h1

Beginner2022-04-23Added 13 answers

Step 1
Let $z (x) = \frac{1}{y (x)}$ ,
so $y^{'} (x) = - \frac{z^{'} (x)}{z {(x)}^{2}}$ ,
Step 2
hence $- \frac{z' (x)}{z (x)^{2}} - \frac{1}{x z (x)} + 3 z (x) (\frac{1}{2} - x) + 1 = 0$ ,
so $z^{'} (x) + \frac{z (x)}{x} + 3 z (x)^{3} (x - \frac{1}{2}) - z (x)^{2} = 0$ ,
which can be further rewritten as
$z' (x) = 3 (\frac{1}{2} - x) z (x)^{3} + z (x)^{2} - \frac{1}{x} z (x)$ ,
which is an Abel equation.

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